Turbulence, integral scale

The turbulence integral scales L and L22 are proportional to a turbulence integral scale Lu defined in terms of k and e ... 

Likewise, the ratio of the transverse Taylor microscale and the turbulence integral scale can be expressed as... 

In a fully developed turbulent flow, the rate at which the size of a scalar eddy of length l,P decreases depends on its size relative to the turbulence integral scale L and the Kolmogorov scale ij. For scalar eddies in the inertial sub-range (ij < Ip, < Lu), the scalar mixing rate can be approximated by the inverse of the spectral transfer time scale defined in (2.68), p. 42 8... 

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

In many reacting flows, the reactants are introduced into the reactor with an integral scale L that is significantly different from the turbulence integral scale Lu. For example, in a CSTR, Lu is determined primarily by the actions of the impeller. However, is fixed by the feed tube diameter and feed flow rate. Thus, near the feed point the scalar energy spectrum will not be in equilibrium with the velocity spectrum. A relaxation period of duration on the order of xu is required before equilibrium is attained. In a reacting flow, because the relaxation period is relatively long, most of the fast chemical reactions can occur before the equilibrium model, (4.93), is applicable. 

The Taylor h rpothesis has been further discussed by [144] (pp 5-7) and [59] (pp. 31-81) considering theories of double correlations between turbulence-velocity components, the features of the double longitudinal and lateral correlations in homogeneous turbulence, integral scales of turbulence, and Eulerian, Lagrangian and mixed Eulerian-Lagrangian correlations. 

For the species concentration field, the scalar integral scale and the Batchelor scale characterize respectively the largest and the smallest eddies. In most cases, the scalar integral scale is approximately equal to the turbulence integral scale. The Batchelor length scale, on the other hand, is related to the Kolmogorov length scale via the Schmidt number, Sc=nlpD ... 

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